More Thoughts on Problem Solving

We have finished up our second group project focused on using problem solving to teach mathematics. As we wound it up we were presented with a couple of thoughts that, I think, show us where we are today in our understanding of such a task.

The first thought was that, in 1980, NCTM released a document titled 'An Agenda for Action: Recommendations for School Mathematics of the 1980s'. Akihiko Takahashi, our instructor, discussed with us a couple of the phrases from that document on how problem solving should be used in a mathematics classroom. He then noted that a major reform in Japanese mathematics teaching took place as a result of this document. They, the Japanese, took what was said and began implementing it. Today they are seen as leaders in the teaching of mathematics in their k-12 schools. What about those of us in the U.S? Where are we? It seems that we missed out on the general ideas of this document. It was our organization that crafted this document, but somehow, it seems, it did not lead to widespread change in the teaching of mathematics. I may be seeing it wrong, but from what I have read, studied, and experienced, I think I am right in my assessment.

The second thought I had relates to teachers working together to create problems and lessons that will help students learn mathematics through problem solving. As we struggled with each other, or more accurately against each other, while trying to put two such lessons together in two weeks, there were a lot of groups that struggled to work well together. We were told that we needed to spend more time learning each other, rather than trying to teach each other. Apparently we were busy trying to tell each other the way it should be and why other ideas would not work.

The last thought I had came as we discussed why we had been given such a task. Aki reminded us that we are often bombarded with great activities and problems at conferences, but never take the time to figure out what mathematics the students could learn from such an activity. We simply go and use the activity as is, and as a result we often have a less than desired outcome. In short, we need practice turning good problems into tasks where students learn mathematics.

Finally, although at times I was frustrated with these tasks and with the dynamics of the groups I was working with, I did learn a few things that I would like to implement in my teaching. I do think that mathematics as is commonly taught is more boring than it should be. I hope to figure out some of this stuff in time to try a few things in the classroom this year, and then record what happened and use that to improve in the coming years.

How the World Really Works

2006 Fields Medalist Andrei Okounkov discussed the Law of Large Numbers and a few of its consequences. Essentially if we are looking at a random path of n steps that starts at ordered pair (0,0) and goes to (T,X), where T is some time and X is some location, then as the number of steps gets very large the probability that the path varies from a straight line is next to nothing. In other words by far the most likely thing to happen is a straight line between (0,0) and (T,X), which turns out to not be that random after all.

This result was extended to cover three dimensional walks as well. What this means, at least as far as I understand it, is that the things we see as very precise and not random at all are actually so random that they gain, as Okounkov said, an Optimality and Elegance that makes us think they are not random.

All in all I felt it was one of the most understandable of the Clay Lectures I have attended in my two years at PCMI. The speaker was funny, and to my surprise, a little bit uncomfortable at times. This seemed to be in part because English is not his native language and in part because of the challenges of addressing such a diverse crowd. He took the time to state things in a way that was very accessible, which was nice. Having the chance to interact with such individuals during mealtimes and in such intimate settings is a feature that I have only experienced at PCMI.

And so we see that our world is, in fact, very random.

Problem Solving FOR Learning

Today we finished discussing an activity we created last week. We were to take a paper fold and use it to teach mathematics. The most challenging aspect of this task was to design a problem that would actually teach a mathematical concept. I, and it seemed like many of the others, have traditionally used problem solving activities as a means to test our students' knowledge about a particular topic.

Several times in our discussions we were asked "What new math are they learning as a result of this problem"? This struck me as an important aspect to teaching mathematics that I have overlooked. One portion of what we do as teachers at PCMI is to learn more mathematics. In these lessons I find it much more interesting to explore things that I don't necessarily know the formula for. I think that, with some training, our students will be the same. They will be much more interested in learning the math when the problems they are solving are teaching them this math.

I will let you know when I have it figured out.

iTunes Data in Fathom

This afternoon in our PCMI SSTP (Secondary School Teachers Program) data working group we had the chance to mess around with Fathom. I have not been overly impressed to this point with the friendliness of the Fathom interface, but am getting more comfortable as I play with it. I am not convinced that I will be letting my students touch it anytime soon, but as the start of the school year is still two months away, that is ok.

I was wondering how to spend the two hours of Fathom-time and I decided to see if I could import the data from my iTunes library. I had to fiddle with things a bit, but after

1. choosing export from the iTunes File menu (not Export Library...), and then
   
2. importing this .txt file into an Excel spreadsheet,
   
3. cleaning up the data a bit,
   
4. exporting it as tab delineated .txt file, and then
   
5. importing this file into Fathom I had my data in the form of a collection.

I created a box plot that had Genre as the y-axis and Play Count as the x-axis. From this I determined that my country music songs are played more often than the other genres that I had. A histogram showed that the number of country songs in my iTunes is larger than the number of songs from any other genre. A dot plot with file size on the y-axis and time of song (which shows in seconds) on the x-axis showed a linear pattern with very few files outside the lines. The files with larger file sizes relative to the length of the song were files of the MPEG audio format. This makes some good sense, since the default in iTunes is to import songs in the more compressed AAC format.

It wouldn't be too much trouble to design a lesson where we compiled students' iTunes library information and looked for trends among classes or found the equation of the line that fits the dot plot of song length vs. file size. I am sure I will get time to look at this over the course of the next few weeks.

Governor Huntsman Welcomes PCMI Participants

This morning we heard from Governor Huntsman (Utah). He welcomed participants of PCMI (Park City Math Institute) to Utah, and asked us to consider several things as we participate in the three week institute. Here are a few of his thoughts and questions (paraphrased) and some of my responses.

• After the first three years of teaching, 50% of the math teachers leave the classroom. There are many pursuits professionally that lure them away, but we need to get something figured out to keep them. How do we do this?

I think that a major step in keeping teachers will be to pay them competitive salaries. I know there are a lot of people out there who say that teachers only work 9 months of the year, so why should we pay them more. I agree. I propose that new teachers be required to work 12 months of the year, and that starting salaries for this 12 month work schedule be double the current starting salary (in my district this would mean approximately $56,000). Teachers would spend their summers working on curriculum, technology implementation, and/or improving their teaching methods.

• There should be a concern for the new underclass of mathematically and scientifically illiterate. How is it that we begin to close the gap as kids work through their early years of schooling so that we don’t develop the underclass that could exist. Teachers are a key to inspiring our kids to pursue math and science.

One of the reasons that students don't pursue mathematics and sciences is that they don't have the opportunity to experience math and science in the secondary school setting. Allowing students to get into the mathematics on their own is necessary to getting them interested in pursuing math long enough to gain numeracy.

• We are entering a STAR (Science, Technology, and Research) world. How best can we reach back into our high school and middle schools to help them meet the bench marks of literacy and numeracy that will prepare them for this STAR world?
  

This relates to the first two problems I have mentioned. We need to pay quality teachers, as they are the key to inspiring our students to pursue math, science, and technology. We need to require our new teachers to learn to teach better. As one of the presenters today pointed out, teaching is a profession that needs to be learned. We all benefit from practice. As we require teachers to spend their summers improving their teaching, their teaching will improve.

And finally he gave us a homework assignment, which is essentially to answer the questions and problems posed above. Here are the questions as I wrote them this morning.

How do we shift the focus to numeracy? As we move toward heightening literacy standards, what do we do for numeracy? What benchmarks should we have? What do we do for the teachers? What nationally do we look at? What do we do to keep math teachers in place (differential pay, something else)?